Cauchy problem for NLKG in modulation spaces with noninteger powers

TL;DR

This paper investigates the well-posedness of the nonlinear Klein-Gordon equation with non-integer power nonlinearities in modulation spaces, demonstrating advantages in both high and low regularity regimes.

Contribution

It introduces a novel approach to handle non-integer power nonlinearities in modulation spaces and extends the analysis to the Klein-Gordon-Hartree equation.

Findings

Established local well-posedness in modulation spaces for non-integer powers

Proved global existence for small initial data in Klein-Gordon-Hartree equation

Highlighted advantages of modulation spaces in various regularity settings

Abstract

In this paper, we consider the Cauchy problem for the nonlinear Klein-Gordon equation whose nonlinearity is $|u|^{k}u$ in the modulation space, where $k$ is not an integer. Our method can be applied to other equations whose nonlinear parts have regularity estimates. We also study the global solution with small initial value for the Klein-Gordon-Hartree equation. By this we can show some advantages of modulation spaces both in high and low regularity cases.